A theory of state-to-state transitions based on the framework of classical reaction dynamics

TL;DR

This work develops a Liouville-equation–based theory for state-to-state transitions within classical reaction dynamics, introducing a Markov-boundary-crossing approximation to derive tractable integral equations for state populations $P_j(t)$. By coupling this coarse-grained dynamics to returning-probability (RP) theory, the authors show how binding and unbinding kinetics can be computed from short-timescale MD trajectories, with long-timescale behavior recovered through a set of local, few-state interactions. The method yields unbinding time constants $ au_{ m off}$ in close agreement with brute-force MD across three systems, and reproduces binding rate rankings with reasonable accuracy (within ~40%), while offering substantial reductions in required trajectory length, especially for slow crown-ether/K$^+$ binding. The approach promises practical impact for studying complex biomolecular binding processes and can be extended with techniques such as MMVT to further improve sampling efficiency.

Abstract

We propose a new method to describe the population dynamics of distinct configurational states based on a continuous-time description of state-to-state transitions. According to classical reaction dynamics theory, the probability density associated with a given state obeys the Liouville equation, the probability density associate d with a given state obeys the Li ouville equation, including influx from and efflux to neighboring states. By introducing a Markov approximation for the crossing of boundaries separating the states, tractable integral equations governing the state populations are derived. Once the time-dependent quantities appearing in these equations are evaluated, the population dynamics on long timescales can be obtained. Because these quantities depend only on a few states in the local neighborhood of a given state, they can be computed using a set of short-timescale molecular dynamics (MD) simulations. We apply the present method to the binding and unbinding kinetics of CH$_4$/CH$_4$, Na$^+$/Cl$^-$, and 18-crown-6-ether (crown ether)/K$^+$ in water. For both kinetics, the time constants estimated from the present method are almost comparable to those obtained from brute-force MD simulations. The required timescale of each MD trajectory in the present method is approximately two orders of magnitude shorter than that in the brute-force MD approach in the crown ether/K$^+$ system. This reduction in the trajectory timescale enables applications to complex binding and unbinding sy stems whose characteristic timescales a re far beyond those directly acce ssible by brute-force MD simulati ons.

Figures (7)

• Figure 1: Reaction-coordinate space and its division into a set of states, $\bm{\Upsilon}_{1},\bm{\Upsilon}_{2},\cdots$. The normal vector on the boundary between states $i$ and $j$ is defined as ${\bf n}_{ij}$, where the vector points toward state $i$.$s_{ij}$ signifies the boundary separating states $i$ and $j$.
• Figure 2: Schemes of the governing equations for (a) $P_{j}\left(t\right)$, (b) $Q_{ij}\left(t\right)$, and (c) procedure for computing $P_{j}\left(t\right)$ from MD simulations.
• Figure 3: Boundary conditions used for computing (a) $k_{\mathrm{ins}}$ and (b) $P_{\mathrm{RET}}\left(t\right)$. $\mathcal{M}_{\mathrm{absorb}}$, $\mathcal{M}_{\mathrm{reflect}}$, and $\mathcal{M}_{\mathrm{R}}$ denote the sets of absorbing, reflecting, and reactive states, respectively.
• Figure 4: Target binding/unbinding systems: (a) CH$_4$/CH$_4$, (b) Na$^+$/Cl$^-$, and (c) crown ether/K$^+$, each immersed in pure water. The molecular structures are visualized using Visual Molecular Dynamics (VMD) package.humphrey1996vmd
• Figure 5: Potentials of mean force (PMFs), $w\left(r\right)$, for (a) CH$_4$/CH$_4$, (b) Na$^+$/Cl$^-$, and (c) crown ether/K$^+$. Bound region (highlighted in gray) is defined by the barrier top position.